Characterization and Performance Optimization of 2D Lattice Materials With Hexagonal Bravais Lattice Symmetry

Abstract

The current paper examines the static performance of 2D infinite lattice materials with hexagonal Bravais lattice symmetry. Two novel microscopic cell topologies are proposed. The first topology is a semi-regular lattice that has the modified Schafli symbol 34.6, which describes the type of regular polygons surrounding the joints of the lattice. Here, 34.6 indicates four (4) regular triangles (3) successively surrounding a node followed by a regular hexagon (6). The second topology is an irregular lattice that is referred here as Double Hexagonal Triangulation (DHT). The lattice material is considered as a pin-jointed micro-truss where determinacy analysis of the material micro structure is used to distinguish between bending dominated and stretching dominated behaviours. The finite structural performance of unit cells of the proposed topologies is assessed by the matrix methods of linear algebra. The Dummy Node Hypothesis is developed to generalize the analysis to tackle any lattice topology. Collapse mechanisms and states of self-stress are deduced from the four fundamental subspaces of the kinematic and the equilibrium matrices of the finite unit cell structures, respectively. The generated finite structural matrices are employed to analyze the infinite structural performance of the lattice using the Bloch’s theorem. To find macroscopic strain fields generated by periodic mechanisms, the Cauchy-Born hypothesis is adopted. An explicit expression of the microscopic cell element deformations in terms of the macroscopic strain field is generated which is employed to derive the strain energy density of the lattice material. Finally, the strain energy density is used to derive the material macroscopic stiffness properties. The results showed that the proposed lattice topologies can support all macroscopic strain fields. Their stiffness properties are compared with those of lattice materials with hexagonal Bravais symmetry available in literature. The comparison showed that the lattice material with 34.6 cell topology has superior isotropic stiffness properties. When compared with the Kagome’ lattice, the 34.6 lattice generates isotropic stiffness properties, with additional stiffness to mass ratio of 18.5% and 93.2% in the direct and the coupled direct stiffness, respectively. However, it generates reduced shear stiffness to mass ratio by 18.8%.